Help with multilinear forms quesion - Linear algebra

[tight haven]

What does the matrix A look like?

[willow crown]

Sorry about that should’ve made it more clear. It’s only s2 I’m stuck on which doesn’t involve A

[tight haven]

Okay, if I tell you that $A : \mathbb{R} \to \mathbb{R}$ is a linear map and $A(1) = 2$, can you tell me what $A(2)$ would be?

[grizzled dagger]

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[willow crown]

No I don’t see how I could without knowing what A is or with more entries. Maybe my understanding of linear maps is not great

Would it help at all to write the vectors in the problem in terms of the basis vector of R^2

[tight haven]

Yes, that would be a good first step

[willow crown]

You said to write x,y,z in terms of w1, w2, w3 but I’m not sure how this helps to begin with

[tight haven]

Well...

I see that the obvious way to do it is to solve the 4 linear equations for things like $f(e_1, e_1, e_1), f(e_2, e_2, e_2), f(e_1, e_1, e_2), f(e_1, e_2, e_2)$, where $e_1, e_2$ are the standard basis vectors.

[grizzled dagger]

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[tight haven]

You have four equations for these four unknowns given by $f(v_i, v_i, v_i) = r_i$ for $i = 1, 2, 3, 4$.

[grizzled dagger]

11de784a

[tight haven]

There is almost definitely a cleverer way, but it is 30 minutes past my bedtime.

[tight haven]

Once you have these, you can use trilinearity and symmetry to evaluate any $f(x, y, z)$.

[grizzled dagger]

11de784a